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Speeding up the spatial adiabatic passage of matter waves in optical microtraps by

optimal control

Antonio Negretti1, Albert Benseny2, Jordi Mompart2, and Tommaso Calarco11Institut für Quanteninformationsverarbeitung, Universität Ulm, D-89069 Ulm, Germany and

2Grup d’Òptica, Departament de F́ısica, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain

(Dated: November 10, 2018)

We numerically investigate the performance of atomic transport in optical microtraps via the socalled spatial adiabatic passage technique. Our analysis is carried out by means of optimal controlmethods, which enable us to determine suitable transport control pulses. We investigate the ultimatelimits of the optimal control in speeding up the transport process in a triple well configuration forboth a single atomic wave packet and a Bose-Einstein condensate within a regime of experimentalparameters achievable with current optical technology.

PACS numbers: 03.67.Lx,34.50.s,

I. INTRODUCTION

The coherent control of matter waves has become avery relevant research topic with significant technologi-cal applications such as in atom lasers [1–3] and quan-tum information processing (QIP) [4], to only cite a few.Indeed, very recently, experimental demonstrations withultracold atoms in two-dimensional (2D) optical lattices,of single quantum bit (qubit) rotations [5], single siteaddressability [6, 7], and single atom detection [8, 9],with very high fidelities, have been reported. One ofthe current most important goals of QIP is to go be-yond the manipulation of a handful of qubits since themain challenge is to build scalable quantum hardwarewhere several thousands of qubits are coherently manip-ulated within the relaxation times of the system [10].It has been also understood that the practical realiza-tion of QIP requires to devise new architectures. Indeed,several schemes for two-qubit quantum gates implemen-tations based on atomic systems proposed in the pastcan be, at least in principle, realized in experiments, butthey may present several shortcomings when building ascalable quantum computing hardware (see Ref. [11] fora review on QIP with neutral particles). Moreover, inorder to name a QIP system “scalable”, it is also impor-tant that the resources required to control the quantumsystem, typically classical devices (e.g., laser fields, re-frigerators), are scalable as well [12].

Paradigmatic examples of QIP are the Cirac-Zoller [13]and the Mølmer-Sørensen [14] ion quantum computers,which have been proven to be powerful schemes to ex-perimentally realize small quantum algorithms [15–17]or to engineer quite exotic entangled states (up to 14qubits), like Werner [18] or Greenberger-Horne-Zeilingerstates [19, 20]. These schemes, however, present ratherdifficult technical problems when hundreds or even thou-sands of ions participate in the collective motion (e.g,decoherence of motional modes, sensitivity to electricnoise [21]). Thus, currently, a big effort is made in thedesign and practical realization of new schemes that areactually scalable. For instance, in Ref. [22] it has been

proposed an ion quantum processor architecture, wheresome areas of the chip processor are used only to storethe information, and others to manipulate it. Such adesign requires to transport an ion from one locationto another one in the chip preserving its quantum me-chanical coherence. A similar problem is encountered forQIP implementations with neutral atoms either in opti-cal lattices [23–25] or in microwave atom chips [26, 27].Atoms trapped in optical lattices can be efficiently pre-pared via the superfluid Mott insulator quantum phasetransition [28, 29], and single sites addressed [6, 7], butthe realization of quantum gates between qubits locatedin far away lattice sites can be a serious problem for ascalable neutral-atom-based quantum processor. Solu-tions to this issue can be afforded, for instance, by aux-iliary atoms that can be efficiently transported in state-independent periodic external traps [30] or by using op-tical tweezers [31].A major underlying concept of these QIP paradigms

is the coherent transport of ions or atoms in such a waythat they mediate the operation of quantum gates be-tween spatially distant qubits. The needed transporttime, however, has been estimated to be about 95% ofthe time used for carrying out the whole quantum com-putation [32]. It is therefore imperative to reduce thetime needed to transport an atom or ion from the quan-tum memory to the processing units and, therefore, toengineer robust control transport pulses. In this respect,optimal control theory is a prominent candidate for adrastic improvement of the design of accurate QIP pro-tocols, and, recently, several theoretical investigationson the optimal transport of both a single atom and anatomic ensemble have been undertaken [33–36]. Besidesthis, very recently, control pulses numerically obtainedby using iterative optimization algorithms have been ex-perimentally applied, with great success, in order to effi-ciently transfer a one-dimensional (1D) degenerate Bosegas from the transverse ground to the lower excited stateof a waveguide potential [37]. This result shows the po-tential offered by optimal control methods to engineercurrent experiments of ultra-cold atoms.In this work we investigate the ultimate limits of the

http://arxiv.org/abs/1112.3828v1

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transport of neutral atoms in optical microtrap arrays bymeans of numerical optimization methods. Specifically,we are interested in the spatial adiabatic passage (SAP)protocol [38], the matter wave analogue of the stimulatedRaman adiabatic passage (STIRAP) technique used inquantum optics to transfer the population between twoatomic internal levels [39]. The SAP technique consistsin adiabatically following an energy eigenstate of the sys-tem, the so-called spatial dark state, that only involvesthe vibrational ground states of the two extreme wellsof a triple-well potential (see Fig. 1). The spatial darkstate presents at all times a node in the central regionsuch that the middle-well population is almost negligi-ble throughout the transport process. Therefore, theSAP protocol enables to transport an atom from a lat-tice site to the next-nearest-neighbor without populatingthe nearest-neighbor site (the middle well in Fig. 1). Ifone atom is present in the middle trap, the SAP tech-nique can be used to implement a single atom diode ora transistor [40]. In addition, this transport techniquemay allow to reduce the complexity of several quantumcomputing architectures [7, 26, 41–43]. In fact, comparedto the tunneling induced oscillation between two adjacenttraps, such technique is more robust against variations ofthe system parameters and requires less precise controlof the distance and timing. Three-level atom optics tech-niques [38], such as the SAP protocol, allow also to createsuperpositions (spatial dark states) of matter waves be-tween two separated sites of the lattice, useful for appli-cations in atomic interferometry, or to inhibit the tunnel-ing among lattice sites, therefore allowing to create con-ditional phase shifts for quantum logic, or to transportan empty site [38]. We also mention, that, recently, theimplementation of the SAP protocol for radio-frequencytraps [44–46] within the three mode approximation hasbeen investigated [47], and that such technique could beeven employed for the ion transport in segmented micro-traps [48, 49].

Beside this, we will investigate the transport of a Bose-Einstein condensate (BEC) in relation to recent experi-ments with optical dipole traps [50–52]. We note that asimilar study has been carried out in Ref. [33], where theoptimal transport of a BEC in magnetic microtraps, likethe ones produced with atom chips [4], has been investi-gated, and that, very recently, the optimal control pulsesfor harmonically trapped BECs have been analyticallydetermined [36]. We underscore that, while the goal ofthose investigations was to transfer a BEC between spa-tially separated locations, here, in addition to this goal,we aim at minimizing the occupancy of the middle wellin a triple-well configuration, as showed in Fig. 1, byfollowing as much as possible the spatial dark state ofthe trapping potential. This additional constraint is themain signature of the SAP protocol.

−10 −8 −6 −4 −2 0 2 4 6 8 100

1

2

3

V(x

)[u

nits

of

hν x

]

φ(1)− 1(x) φ

(2)− 1(x)

−6 −4 −2 0 2 4 60

1

2

3

x [µm]

V(x

)[u

nits

of

hν x

]

φ(2)− 1(x) φ

(2)1 (x)

Figure 1. (Color online). Upper panel: Initial (black thickline) and final (blue thin line) potential configuration for step1 of the transport process with trap separation 6.5 µm, beamwaist 2w = 1.3µm, potential depth V0 = kB × 86 nK, andtrap frequency ωx =

√

4V0/mσ2 = 2πνx ≃ 2π × 711 Hz.

The initial ground state wave function ψ0(x) ≡ φ(1)−1(x) (red)

of the left well is also displayed. Lower panel: Initial andfinal potential configuration (same as the blue thin line ofthe upper panel) for the SAP process. The goal wave func-

tion ψ′g(x) ≡ φ(2)−1(x) (bright grey) of step 1 is shown. This

state also corresponds to the initial wave function of the SAP

process, whose goal wave function is ψg(x) ≡ φ(2)1 (x) (dark

green), superimposed on the right well of the lower panel.The horizontal black arrows indicate the transport direction.

II. OPTICAL DIPOLE MICROTRAPS

We consider a (transverse) potential, where either asingle atom or BEC is trapped, given by the followinganalytical expression

V (x, t) = V0

{

1−1∑

k=−1

vk(t) exp

[

− (x− kdk(t))2

2w2

]

}

,

(1)where V0 represents the depth of the three Gaussiandipole traps, with 2w being the laser beam waist, d−1(t)represents the distance between the central trap and theleft trap, d1(t) is the distance between the central trapand the right trap, while the central trap remains atd0(t) ≡ 0 ∀t ∈ [0, T ]. Here T is the time needed totransport the system of interest (i.e., an atom or a BEC)initially prepared in the ground state of the trap on the

3

left (centered in −|x′0|) to the ground state of the trapon the right (centered in |x′0|). As outlined above, weshall first consider a 1D scenario, but later we shall alsostudy the influence of the dimensionality on the transportperformance. We note that optical dipole traps, as theones of Ref. [51], can be designed to form a 2D lattice.The x − y plane, that defines the lattice, has a weakerconfinement than the (vertical) axial direction, therebydefining a “pancake” geometry. It is in the (transverse)x− y plane that the transport occurs and because of thiswe refer to the potential (1) along x as transverse (in ourstudy the trap frequencies in the x and y directions willbe assumed to be equal).

In Fig. 1 the potential is illustrated, where experimen-tally realistic parameters (i.e., potential depth, trap sep-aration, and beam waist) have been considered for 85Rbatoms. Due to the large (initial) separation between thetrap minima of the lattice [x′0 = 6.5µm, see Fig. 1 (top)],no tunneling is expected to happen until the traps arecloser. The SAP transport is split and optimized in threedifferent stages. Firstly, in a time T1, the initial atomicstate (the ground state of the left well) is moved fromx′0 = −6.5µm to x0 = −2.5µm, i.e., from the left well ofthe initial trap configuration (thick black line) given inFig. 1 (top) to the left well of the potential of the upperpanel of Fig. 1 (blue thin line). Secondly, the atomic sys-tem is brought, in a time T2, from the left well centered atx0 = −2.5µm to the right one centered at x0 = 2.5µm(see blue line of the lower panel of Fig. 1). The thirdstep consists in bringing the system from the right wellcentered at x0 = 2.5µm to the right well centered atx′0 = −6.5µm of the potential displayed in Fig. 1 (top),which is equivalent to reversing the first step of the pro-cess. Hence, the total transport time is T = 2T1 + T2. Itis in the second step that the SAP process takes place.Since at the beginning the atoms are quite far apart,the direct application of the SAP technique would sim-ply slow down the whole transport process, because ini-tially no tunneling would take place. Instead, with theabove outlined procedure, the first step can be sped upas much as possible until the so-called quantum speedlimit is reached, that is, the minimum time required toevolve a quantum system from an initial state to an otherorthogonal state [53–55].

Finally, we note that a crucial condition for the re-alization of SAP is that the initial and final states in-volved during the transport process should be in reso-nance. This requirement can be fulfilled by fixing at alltimes the minima of the potential at the same energylevel, through the control of the time-dependent param-eters v−1(t), v1(t) [v0(t) ≡ 1] (see also Fig. 2), as wellas by fixing the maxima of the triple well configurationto the same energy level, through the control of w, thatis, the beam waist (see the appendix for the analyticalexpressions of v±1). The control of the latter, however,would require beam waist values below the actual experi-mentally achievable limit [51], and therefore it will not beconsidered in our study. Hence, in our analysis we have

−4 −2 0 2 40

1

2

3

V(x

)[u

nits

of

hν x

]

x [µm]

Figure 2. (Color online). Potentials at time t = T2/2:the blue (solid) line shows the potential when v±1 are time-dependent, whereas the magenta (dashed) line when v±1 aretime-independent. The potential minima have been lifted byabout 0.33hνx and 0.44hνx, respectively. The symmetry ofthe potentials is due to the fact that, at that time, the twoouter traps are equidistant from the centre x = 0, that is, adistance almost equal to the minimal allowed trap separationδx0 = 1.43µm. The rest of trap parameters are as in Fig. 1.

fixed w to the minimum experimentally achievable value(i.e., 0.65 µm), which enables us to prepare the atoms,before the transport, in a lattice configuration with min-imum periodicity.

III. SINGLE ATOM TRANSPORT

In this section we analyze the transport of a singletrapped atom in the absence of a thermal or quantumbath. The atomic state obeys the Schrödinger equationof motion. In order to speed up the transport process werely on numerical optimization techniques. For the prob-lem considered in this paper, we employed a recently in-troduced optimization method, named the chopped ran-dom basis (CRAB) algorithm [56]. Such a method hasbeen shown to be a powerful tool in order to optimize theclosed dynamics of many-body quantum systems [57] andthe dynamics of light harvesting [58]. Moreover, since theimplementation of the CRAB algorithm does not rely onthe equation of motion that governs the system dynam-ics, there is no need of algorithmic modifications whennonlinear dynamics is regarded (e.g., the dynamics of aBEC), unlike for the monotonically convergent Krotovalgorithm [59–61] or the gradient ascent pulse engineer-ing algorithm [62]. Even though CRAB does not providemonotonic convergence it allows to directly restrict andselect the space of control pulses (e.g., enforcing limitedbandwidth), since it relies on a multi-variable functionminimization that can be performed, for example, viaa direct-search method (e.g., the Nelder–Mead methodas implemented, for instance, in MATLAB). For moredetails on the procedure of the CRAB algorithm imple-

4

mentation and its computational performance we refer toRef. [56].

A. Optimization of step 1 of the transport process

We remind that our goal here is to transport an atominitially prepared in the ground state of the left well ofthe potential displayed in Fig. 1 (top), centered in x′0 =−6.5µm, to the ground state of the left well of the poten-tial shown in Fig. 1 (bottom) blue (solid) line, centered inx0 = −2.5µm. In this case we shall consider the controlpulses to be identical, that is, d−1(t) = d1(t) ≡ d(t).In this section, the objective functional (to be mini-

mized) for the control problem we are interested in canbe identified by the so-called overlap infidelity, namely

I = 1−∣

∣

∣

∣

∫

R

dxψ′∗g (x)ψ(x, T1)

∣

∣

∣

∣

2

. (2)

Here ψ(x, T1) is the wave function at time t = T1 propa-

gated from the initial condition ψ0(x) ≡ φ(1)−1(x) at timet = 0, where φ

(1)−1(x) is the ground state of the left trap

centered at x = x′0 = −6.5µm. The wave function ψ′g(x)is the wave function we aim to achieve in a given time T1,

that is, the ground state φ(2)−1(x) of the left well centered

at x = x0 = −2.5µm. The superscript (j) in φ(j)−1 refersto the two first stages of the transport process. When

j = 1 the state φ(1)−1 corresponds to the ground state of

the left well of the potential (thick black line) illustratedin the upper panel of Fig. 1, while for j = 2 it correspondsto the ground state of the left well of the potential shownin the lower panel of Fig. 1 (blue solid line). The same

applies to φ(j)1 , but for the right wells.

The wells of the upper panel of Fig. 1 (thick and thinlines) are sufficiently deep that the trapping potentialscan be, with good approximation, considered harmonic,but with slightly different trap frequencies. Hence, anexcellent guess control pulse is given by [34]

dho(t) = (x′0 − x0)

{

t

T1+ sin

(

2πt

T1

)[

8π

3(ωxT1)2− 2

3π

]

− sin(

4πt

T1

)[

4π

3(ωxT1)2− 1

12π

]}

− x′0. (3)

For a particle in a moving harmonic potential such a con-trol pulse is optimal, i.e., it yields I = 0, and it is quiterobust against control pulse distortions [34]. Neverthe-less, as Fig. 3 shows (solid line), for our case with Gaus-sian traps, we obtain already a good result for large, butnot adiabatic (i.e., not in the regime ωxT1 ≫ 1), trans-port times.To reduce the infidelity we further optimized the trans-

port process by means of the CRAB algorithm, whichworks as follows: we start with the initial guess given byEq. (3) and we define the new control pulse as d(t) =

2.2 2.6 3 3.4 3.8 4.2 4.6 5 5.410

−4

10−3

10−2

10−1

100

I

T1 [ms]

Figure 3. (Color online). Overlap infidelity vs. transporttime: black (solid) line with the control pulse dho(t) definedin Eq. (3); the other two lines show the infidelity obtainedwith the CRAB optimized control pulse dopt(t) = dho(t)gopt(t)for Ng = 8 (dashed line) and Ng = 16 (dot-dashed line).

dho(t)g(t), where

g(t) = 1 +1

λ(t)

Ng∑

k=1

[Ak sin (ωkt) +Bk cos (ωkt)] . (4)

Here ωk = 2πk/T1, Ng ∈ N is the number of time-independent Ak and Bk coefficients, λ(t) is a time-dependent function enforcing the boundary conditionsof d(t) at t = 0 and t = T1, namely limt→0,T1 λ(t) =+∞. Basically, the CRAB algorithm seeks for the time-independent coefficients Ak, Bk and frequencies ωk thatminimize the overlap infidelity (2). Besides, in the nu-merical simulations, we set a tolerance (∼ 10−4) on thedetermination of either the coefficients or the frequen-cies ωk. Such a tolerance is defined as the minimumallowed distance between the vertexes of the polytopegenerated within the Nelder-Mead multidimensional non-linear minimization procedure [63].As illustrated in Fig. 3, the CRAB algorithm slightly

improves the result for large T1 times obtained with thecontrol pulse defined in Eq. (3), but for short times theslight difference due to the proximity of the central trapin the trap frequencies (of about 0.7%) of the left wells[thick and thin lines of Fig. 1 (top)], becomes crucialas well as the anharmonicity of the confinement poten-tial. We have also investigated the improvement of theoverlap infidelity due to a higher number of harmonicsNg involved in the control pulse. As Fig. 3 shows, asignificant improvement can be observed only for initiallarge overlap infidelities, whereas for already almost per-fect transport processes the reduction is almost insignifi-cant. This numerical observation is not surprising, sincethe dynamics with an initial large overlap infidelity re-quire more sophisticated control pulses (i.e., higher har-monics), in order to properly steer the atomic dynam-ics. Such more complex control pulses, however, might

5

0 0.5 1 1.5 2 2.5 32

3

4

5

6

7

d(t

)[µ

m]

t [ms]

Figure 4. (Color online). Optimal control pulses of step 1 ofthe transport process (red dashed line with Ng = 8, blue solidline Ng = 16); initial guess given by Eq. (3) (black thick solidline). The transport time is T1 = 3.4 ms.

be more difficult to implement experimentally. Besidethis, as shown in Fig. 3, we also note that the overlapinfidelity drops quite significantly, as a rule of thumb,for times larger than 3 ms, which is consistent with thefact that the transport time cannot be shorter than theinverse of the typical trap frequency, that is, 1/νx ∼ 2ms. We underscore that the threshold ν−1x cannot beprecisely identified with the quantum speed limit, but itis in close relation with it. An exact determination ofthe quantum speed limit relies on the time average ofthe instantaneous energy fluctuations, which depend onthe particular control pulse. This is non trivial compu-tational task for time-dependent Hamiltonians. Only insimple cases, such as the Landau-Zener model, the min-imum time can be efficiently estimated [55]. However,further optimization, that is, larger values of Ng, willnot overcome this (physical) limit.As an example of the optimization, Fig. 4 shows the

optimized transport control pulses for two different setsof coefficients {Ak, Bk} for the transport time T1 = 3.4ms, where dho(t) has been used as initial guess (blackthick line). We see that, to achieve very small infideli-ties, the control pulse involves more wiggles, especiallyof large amplitude at the intermediate times, where thesystem is more excited. On the other hand, at the endof the transport process the system has to be restoredagain in the ground state of the trap, and therefore themodulation of the control pulse is more “gentle”.We have also investigated the robustness against trap

position fluctuations due to possible experimental imper-fections. To this aim we used for the time-dependentcontrol pulses the following expression

d∓1(t) = dopt(t)± ashake sin(ωshaket), (5)

where dopt(t) is the optimal control pulse obtained withCRAB. With such a choice, for ashake > 0 the left andright wells oscillate in phase, whereas for ashake < 0 their

0 0.1 0.2 0.3 0.4

10−3

10−2

a shake [µm]

I

Figure 5. (Color online). Overlap infidelity vs. amplitude of ashaking in the trap positions of the outer traps with ωshake =10−2ωx, δx0 = 1.43µm, and ℓ =

√

~/mωx = 0.41µm. Thetransport time is T1 = 3.4 ms and Ng = 16.

oscillations are out of phase. However, since in this par-ticular step of the transport process only a populatedwell is effectively involved, the only value that mattersis |ashake| (for negative values the behavior of I is basi-cally the same). The result of such analysis is shown inFig. 5 for ωshake/ωx = 10

−2 and T1 = 3.4 ms (see alsoFig. 3). We see that the optimal control pulse is quiterobust against fluctuations of the trap position. This re-sult is in agreement with the findings for a particle in amoving harmonic potential [34].Finally, we investigated the role of spatial dimension-

ality. Up to now, we performed our analyses in the quasi-1D regime. However, we recall that in the experimentsof Refs. [50, 51] the potential in the z (axial) directionis shallower than in x or y. We therefore performed nu-merical simulations of the 2D Schrödinger equation withthe trapping potential

V (x, z, t) = V0

{

1−1∑

k=−1

vk(t)e−

[x−kd(t)]2

2w2 e− z

2

2w2z

}

.(6)

Here 2wz is the beam waist along the z direction. Theratio ωz/ωx is determined by wx/wz, namely, the largerwz is, the smaller the (axial) frequency ωz. The new(ground) initial and goal states have been obtained by us-ing the imaginary time propagation procedure, typicallyused for the determination of the ground state of a BEC.As trial functions for the imaginary time propagation wetook the tensor product of the solutions of the quasi-1Dregime: for the transverse direction by a numerical exactdiagonalization of the single particle Hamiltonian, andfor the axial direction by choosing the Gaussian groundstate of the harmonic oscillator.The result of such analysis, for the optimal control

pulse obtained for the transport time T1 = 3.4 ms withNg = 16, is shown in Fig. 6. As it is illustrated,the smaller ωz is, the better the infidelity. This be-havior is reasonable: since wz ≫ wx, we can write

6

10−1

100

10−2

10−1

I2

D

ω z/ω x

Figure 6. (Color online). Overlap infidelity vs. the ratioωz/ωx. Here I2D is simply the generalization of Eq. (2) witha two-variable integration. The transport time is T1 = 3.4 msand Ng = 16.

exp(−z2/2w2z) ≃ 1 − z2/2w2z , which implies an almostperfect harmonic potential in the axial direction, therebyalmost separable from the one in the transverse direction.The figure shows, however, that the infidelity becomeslarger than the one obtained in the quasi-1D regime,which is almost one order of magnitude smaller (seeFig. 3). We attribute this enhancement to the not com-pletely negligible coupling between the axial and trans-verse motion. Thus, for an exact infidelity and controlpulse evaluation, a 2D optimization should be performed.Of course, the result relies on the particular chosen ωz/ωxratio, which ultimately is determined by the experimen-tal setup. As already pointed out in Ref. [26], care has tobe taken when calculations with realistic parameters inthe quasi-1D regime are considered. The quantum speedlimit behavior, however, remains fundamentally the sameas the one of the quasi-1D regime outlined above, and thecontrol pulses obtained in this regime would be excellentinitial guesses for the 2D optimization.

B. Optimization of step 2: SAP process

The transport via SAP is the second step of the opti-mization process outlined above, where the initial condi-

tion at time t = 0 is given by ψ0(x) ≡ φ(2)−1(x), whereasψg(x) is the ground state wave function φ

(2)1 (x) of the

right well centered at x = x0 (see Fig. 1).

We start the optimization by using the following initialguess control pulses:

d01(t) =

{

(x0 − δx0) cos2(

πtT2−td

)

+ δx0 t ∈ [0, T2 − td]x0 t ∈ (T2 − td, T2]

(7)

d0−1(t) =

{

x0 t ∈ [0, td](x0 − δx0) cos2

(

π(T2−t)T2−td

)

+ δx0 t ∈ (td, T2].(8)

We note that d1(t) is d−1(t) time inverted. We chose thetime delay between the two control pulses as td = 0.17T2,where T2 is the transport time used to carry out the SAPtechnique. Such a choice is due to the analogy betweenSAP and STIRAP. Indeed, as shown in Ref. [39], the timeover which the two control pulses do overlap, has to fulfillthe (adiabatic) criterion

td >10

mint∈[0,T ]{Ω(d(t)/ℓx)}, (9)

where [38]

Ω(d/ℓx)

ωx= 2

d

ℓx

(

e(d/ℓx)2{1 + d[1 − erf(d/ℓx)]/ℓx} − 1√

π[e2(d/ℓx)2 − 1]

)

,

(10)

with ℓx =√

~/mωx. The tunneling “Rabi” frequency de-scribes the coupling between the left and the middle wellsfor d = d0−1 and between the right and the middle wellsfor d = d01. We note, however, that Eq. (10) is only validfor harmonic trapping potentials. For Gaussian traps theactual Rabi frequency must be numerically assessed, butfor an estimation of the time delay, Eq. (10) can be used.As we will discuss at the end of this section about therobustness of SAP against fluctuations of td, the error in-duced by using Eqs. (9,10) is indeed small, and the valueof td used in our analyses is quite reasonable.In this scenario the CRAB optimization works as fol-

lows. For the guess control pulse d0−1(t) we take

d−1(t) =

{

x0 t ∈ [0, td](x0 − δx0) cos2

[

π(T2−t)T2−td

g(t)]

+ δx0, t ∈ (td, T2](11)

where both g(t) and λ(t) are only defined in the timeinterval (td, T2], but the expression remains the one givenin Eq. (4). Such a choice for d−1(t) ensures that it isalways bounded by δx0 and x0. The control pulse d1(t) isthen the time inverse of d−1(t), that is, d1(t) = d−1(T2−t).As outlined previously, here the goal is not only

the minimization of the overlap infidelity I = 1 −|〈ψg|ψ(T2)〉|2, but also the minimization of the occupancyin the middle trap. To this aim, we use the following ob-jective functional, namely the cost function we want tominimize:

J = 1 + wET2

∫ T2

0

dt[

∆E(t) + |ψd(xC , t)|2]

−[

wdT2 − 2td

∫ T2−td

td

dt|〈ψd(t)|ψ(t)〉|2

+ wg|〈ψg|ψ(T2)〉|2]

, (12)

7

where |ψd〉 is the spatial dark state, which corresponds tothe first excited state obtained by diagonalizing at eachtime the single particle Hamiltonian Ĥ(t) = p̂2/2m +V (x, t), ∆E(t) = |E2(t) + E0(t) − 2E1(t)| with En(t)n = 0, 1, 2 the first three eigenvalues at time t of Ĥ(t) (E1is the energy of the spatial dark state), and xC is the po-sition of the minimum of the middle well, where the nodeof the spatial dark state should be located. The energydifference ∆E(t) is used to keep the energy of the spatialdark state equidistant from the energies of the groundand second excited states, and reduce the transitions outof the dark state. The second line of Eq. (12) correspondsto the projection of the evolved state on the actual spa-tial dark state in the interval [td, T2 − td], whereas theweights wE , wg, and wd can be adjusted for convergence.We note that a similar objective functional has been usedin Ref. [64].We first investigate the behavior of the SAP process

without optimization. In Fig. 7 we show the overlap fi-delity F = 1 − I as a function of the transport time T2when the trap parameters v±1(t) are chosen to be time-dependent (blue-bright line), which fixes the minima ofthe triple well potential to the same energy level. Instead,the black-dark line corresponds to the scenario for whichv±1(t) ≡ 1 ∀t ∈ [0, T2]. In this case the first three lowesteigenstates of the Hamiltonian Ĥ(t) are not in resonanceand the evolved state tries to follow the second excitedeigenstate, as also illustrated in Fig. 7 by the behaviorof the overlap infidelity at long times. This phenomenonoccurs because when the two outer wells approach themiddle one, the minima of the outer wells correspondto a larger energy than the minimum of the middle well(see also the magenta dashed line in Fig. 2), and there-fore it is energetically more favorable for the system tofollow the second excited state. Additionally, we notethat fixing the minima of the triple well configuration tothe same energy level yields a spatial dark state whosenode is not localized in the centre of the middle trap, asdesired, but it is lifted towards the outer well with lowerbarrier (or potential maximum). In order to have a nodein the middle trap one should also require maxima at thesame energy level, but this in not contemplated in ourstudy as outlined in Sec. II. Thus, the dark state has tobe properly engineered and this also explains our choicefor the cost functional (12).As shown in Fig. 7 (bright-blue line on the top), the

asymmetry of the potential, due to the fact that the max-ima of the triple-well potential are not fixed to the sameenergy level (while the minima are), does not enable the

atom to follow an eigenstate of Ĥ(t), in particular thespatial dark state, and an oscillatory behavior occurs.Interestingly, the occupancy in the middle trap, definedas

Pc(t) =∫ xmaxR (t)

xmaxL

(t)

dx|ψ(x, t)|2, (13)

Pc(T2) =1

T2

∫ T2

0

dtPc(t), (14)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

F(T

2)

0 0.05 0.10

0.5

1

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

T2 [ s ]

Pc(T

2)

0 0.05 0.10

0.25

0.5

Figure 7. (Color online). Overlap fidelity F = |〈ψg|ψ(T2)〉|2

(top) and probability of occupancy of the middle trap (bot-tom) vs. time. The blue (bright) lines refer to time-dependenttrap parameters v±1(t), whereas the black (dark) lines referto time-independent trap parameters v±1(t) ≡ 1. The in-sets show zooms of both the overlap infidelity (top) and theoccupancy probability (bottom). The diamond and squaresymbols on the top of the blue (bright) line of both panelsrepresent the situations for which the CRAB optimizationhas been performed. In both cases δx0 = 1.43µm.

is higher when we force the minima of the trapping poten-tial to be at the same energy level (i.e., time-dependentv±1). Here x

maxL,R (t) are the positions at time t of the

maxima of the trapping potential of Fig. 1 (lower panel).

In figure 8 we show the optimal control pulses d−1(t)[panel (a)] together with the corresponding probabilitiesof occupancy Pc(t) [panel (c)] for the transport timeT2 = 31.4 ms, that is, the red diamond symbol in theinsets of Fig. 7 (we recall that d1 is the time invertedcontrol pulse of d−1). Instead, in panel (b) the trapparameter v−1(t) is displayed, which corresponds to theoptimal quantum dynamics for T2 = 31.4 ms with time-dependent trap parameters and with objective functionalgiven in Eq. (12) (v1 is basically the time inverted ofv−1). For the CRAB optimization we used Ng = 25 andwe set wg,d = 0.5 and wE = 1. Such a choice is a goodtrade-off between small overlap infidelity and occupancy.In Table I, we show the results of both the overlap infi-delity and probability of occupancy, for which a furtherimprovement was not possible. Indeed, our attempts atoptimizing by considering d±1(t) as independent controlpulses, by looking for an optimal set of frequencies ωk,instead of coefficients {Ak, Bk}, or by varying td, havenot been able to further improve the obtained results.

Given our findings in Fig. 7, we have also analyzed thescenario for which v±1 = 1. In this case we used for the

8

optimization the following objective functional

J = wI[

1− |〈ψg|ψ(T2)〉|2]

+ wPPc(T2), (15)

where the weights wI , wP are adjusted for convergence.The results of such analysis are illustrated in both Table Iand Fig. 8.Concerning the final probability of occupancy in the

middle trap, Pc(T2), the optimization decreases the valuefrom 0.1166, obtained with the initial guess control pulsesd0±1, to the value of 0.0466, obtained with the optimalcontrol pulse, when the system tries to follow the secondexcited eigenstate; from 0.1498 to 0.0771 or to 0.0916 inthe other two cases when the system is forced to followthe spatial dark state (see also Table I). As illustrated inFig. 8 (c), the probability distribution function Pc(t) ispeaked around t = T2/2, that is, when the trap separa-tion is minimum. For comparison, we also show the prob-ability distribution for T2 = 44.8 ms (the green squaresymbol in the insets of Fig. 7), after optimization. In thissituation, the distribution is almost symmetric with re-spect to t = T2/2, because the optimized dynamics tendsto split the transport of the wave packet from the leftto the right well not directly, but in a two-step-like pro-cess, where at time t = T2/2 the state is almost a darkstate. This fact is also confirmed by the pair (pd, nd)of values of the projection onto the instantaneous spa-tial dark state [second line of Eq. (12)] and the node ofthe spatial dark state [second integrand in the first lineof Eq. (12)]: for T2 = 31.4 ms we have (0.211,0.027),whereas for T2 = 44.8 ms we get (0.151,0.011). Thus, forlonger times, the system follows more closely the spatialdark state when the objective functional (12) is chosen.A final remark on the choice of the objective func-

tionals comes from the following fact. As shown in thesecond and third rows of Table I, both the overlap in-fidelity and the probability of occupancy in the middlewell are decreased when the objective functional (15) isused in the CRAB optimization. This choice implies abetter achievement of our goals as well as a computation-ally less demanding simulation. Indeed, with (15) thereis no need to diagonalize the instantaneous single particleHamiltonian Ĥ(t). Hence, with this choice it is possibleto efficiently optimize the transport in optical superlat-tices containing different interacting atomic species [64]and, particularly relevant for our purposes, the transportof a condensate. In this case the determination of the in-stantaneous spatial dark state and its eigenvalue are moreinvolved [65].Finally, as in Sec. III A, we investigated the robustness

of the optimized dynamics against trap position fluctu-ations by adding a shaking term to the control pulsesdopt∓1(t), as in Eq. (5). Moreover, since the CRAB al-gorithm determines the optimal time-independent coef-ficients Ak and Bk, we were also able to investigate theeffect of an imprecise control of the time delay td on theoverlap infidelity. The result of such analysis is illus-trated in Fig. 9 for ωshake/ωx = 10

−2 and for T2 = 31.4ms. We compare the obtained results for two cases: when

T2(ms) I Pc(T2)

31.4⋆ 0.0007 0.0466

31.4† 0.0035 0.0771

31.4 0.0048 0.0916

44.8 0.0028 0.0699

Table I. Optimized errors for different transport times of theSAP technique for a single atom. The first row correspondsto the case for which v±1 = 1, whereas in the other cases v±1are time-dependent. For the first two rows the objective func-tional used in the CRAB optimization is defined in Eq. (15),but for the second row we fixed the minima of the trappingpotential to the same energy level. Instead, for the last tworows, the objective functional is defined in Eq. (12).

the atomic system is forced to follow the instantaneousspatial dark state (right) and when it is forced to fol-low the second excited state of the trapping potential(left). As already pointed out in Ref. [38], for purely har-monic confinement, the SAP process is less affected byimprecise timing, but the overlap infidelity drops fasterfor fluctuations in the trap positions than in the idealcase considered in Ref. [38]. Indeed, while for harmonictraps the fidelity is reduced by 1-2% [38], at the optimaltime delay and ashake ∼ ℓx/2 (ℓx is the harmonic oscilla-tor ground state width), for Gaussian traps the fidelity. 60% (right picture), which shows how detrimental theanharmonicity of the trapping potential is [66]. How-ever, this phenomenon is also due to the fact that thesystem follows the second excited state, which is moresensitive to energy losses, and therefore to a worsening ofthe transport fidelity. Additionally, the figure shows thefragility of the dynamics that the instantaneous spatialdark state of the system is forced to follow, especiallyconcerning the control of the time delay. This furtheranalysis confirms what we already noticed in the overlapinfidelity and occupancy probability, that is, it is moreefficient to follow the second excited state rather thanthe spatial dark one.

IV. TRANSPORT OF A CONDENSATE

In this section we investigate the optimal transport ofa BEC in optical dipole potentials such as the ones inEq. (1). We assume the quasi-1D regime of quantumdegeneracy and a mean field description of the atomicsystem dynamics, that is, we assume that the Gross–Pitaevskii equation (GPE) [67]

−i~∂ψ(x, t)∂t

= Ĥgp[ψ]ψ(x, t),

Ĥgp[ψ] =

[

− ~2

2m

∂2

∂x2+ V (x, t) + g1DN |ψ(x, t)|2

]

(16)

9

0 5 10 15 20 25 301

1.5

2

2.5

3d−

1(t)

[µm

](a )

0 5 10 15 20 25 301

1.1

1.2

v−

1(t)

[units

of

V0] (b )

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t [ms]

Pc(t

)

(c )

0 5 10 15 20 25 30 35 40 450

0.1

0.2

0.3

0.4

t [ms]P

c(t

)

(d )

Figure 8. (Color online). (a) Optimal control pulses obtained by means of the CRAB algorithm respectively for the threesituations listed in the first three rows of Table I: T2 =31.4

⋆ms (blue dashed line), T2 =31.4†ms (black solid line), and T2 =31.4

ms (red solid thick line). (b) Trap parameter v−1(t) for the optimal control pulse T2 =31.4 ms (red solid thick line) of panel(a). (c) Probability of occupancy of the middle well vs. time obtained with the optimal control pulses for T2 =31.4

⋆ms (bluethin line) and T2 =31.4 ms (red thick line). (d) Probability of occupancy of the middle well vs. time obtained with the optimalcontrol pulse for T2 =44.8 ms. Here the time delay is td ≃ 5.2 ms for T2 = 31.4 ms, td ≃ 7.5 ms for T2 = 44.8 ms, and for bothtransport times δx0 = 1.43µm.

well describes the physics of our problem. Here ψis normalized to one, N is the atom number, g1D =

2~ω⊥as3D

(

1− 1.4603as3D

a⊥

)−1

[68], as3D is the three di-

mensional (3D) s-wave scattering length, and a⊥ =(2~/mω⊥)

1/2. This assumption implies that the radialconfinement is frozen to its ground state, and thereforethat the ratio η = ω⊥/ωx is significantly larger than 1(ω⊥ is the transverse trap frequency, that is, the trapfrequency in the y− z plane). As we already pointed outin Ref. [26], a good value is η = 20, for which radial exci-tations due to two-body collisions can be suppressed. Weunderscore that a simulation of the current experimentalsetup would require a 3D simulation of the GPE, sincethe transport occurs in the transverse direction while theaxial confinement has a shallower trap than the trans-verse one, where the transport process occurs.

A. Attractive inter-particle interaction

It is well known (see, for instance, Ref. [67]), that forattractive interactions (i.e., g1D < 0, that is, a

s3D < 0),

a critical number of condensed atoms exists, Ncr, such

that for N > Ncr the condensate is not stable and theGPE has no longer a stationary solution. We have in-vestigated this phenomenon in the quasi 1D regime byconsidering the Gross-Pitaevskii energy functional. Foran arbitrary confinement potential V (x) the functional isdefined as [67]:

E

N=

∫

dx

[

~2

2m

∣

∣

∣

∣

∂ψ(x)

∂x

∣

∣

∣

∣

2

+ V (x)|ψ(x)|2 + g1DN

2|ψ(x)|4

]

.

(17)

For a harmonic trap, a Gaussian Ansatz for the conden-sate wave function can be used, which has been provento provide an excellent estimation of Ncr for a three-dimensional BEC [67]. To this aim, let us consider thefollowing Ansatz for the condensate wave function (nor-malized to 1)

ψ(x) = (σℓx√π)−1/2 exp

(

− x2

2σ2ℓ2x

)

, (18)

where σ is a dimensionless parameter which representsthe effective width of the BEC. By inserting (18) into

10

3.2 4.1 5 5.9 6.8 7.7−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

td [ms]

ash

ake

[µm

]

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

3.2 4.1 5 5.9 6.8 7.7−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

td [ms]

ash

ake

[µm

]

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 9. (Color online). Transport efficiency (i.e., 1 − I) from φ(2)−1(x) to φ

(2)1 (x) as a function of the time delay td between

the two trap approaches and the amplitude ashake of shaking the positions of the outer traps with ωshake = 10−2ωx: (left) for

time-dependent trap parameters v±1, that is, when the system is forced to follow the dark state; (right) for time-independenttrap parameters v±1(t) ≡ 1, that is, when the system is forced to follow the second excited state. Here δx0 = 1.43µm and the“optimal” time delay td ≃ 5.2 ms.

(17) we obtain

E

N~ωx=

1

4

(

1

σ2+ σ2

)

− fcασ√2π, (19)

where f−1c = 1−1.4603as3D/a⊥, and α = ηNas3D/ℓx. Thebehavior of the GP energy functional, for some values ofη and N for an atomic cloud of 85Rb atoms trapped inthe hyperfine state |F = 2,mF = −2〉, is illustrated inFig. 10. As the figure shows, the local minimum disap-pears for either a small atom number (solid vs. dashedlines) or for a small ratio η (solid vs. dashdot lines). Im-portantly, the energy local minimum appears always forE < 0, that is, the interaction energy, Eint, exceeds thekinetic energy. Contrarily to the repulsive case, for at-tractive interatomic forces the kinetic energy, Ekin, can-not be neglected. Indeed, it stabilizes the condensateagainst collapses, namely the condensate is stable as longas Ekin > Eint. We can give a rough estimation of Ncrby using this inequality:

Ekin ∼ −~2

2mℓ2xEint ∼ −2~ω⊥fcN

|as3D|ℓx

, (20)

which imply

Ncr ≃ (4ηfc)−1ℓx

|as3D|. (21)

By fixing η = 20, we have Ncr ≃ 0.3, for ωx = 2π × 711Hz, and Ncr ≃ 2.7, for ωx = 2π × 7.11 Hz. The lattervalue of Ncr could be in principle further enhanced byreducing η, even though the quasi 1D condition wouldno longer be very well fulfilled. Hence, from this analy-sis, we see that, for an attractive BEC in the quasi 1D

0 0.5 1 1.5 2−10

−5

0

5

10E

(σ)/

N[u

nits

of

hν x

]

σ

Figure 10. (Color online). Gross-Pitaevskii energy functionalvs. the effective width in the Gaussian model for attractiveinteracting atoms in the quasi 1D regime. The black (solid)line corresponds to η = 20 and N = 10, the red (dashed) lineto η = 20 and N = 5, and the blue (dashdot) line to η = 5 andN = 10. For all curves we considered 85Rb atoms confined ina harmonic trap with frequency ωx = 2π × 711 Hz.

regime, the admissible condensate atom number is ex-tremely small. With such condensate atomic numbersthe realization of attractive BECs in optical microtrapsis actually not feasible. Given this, the attractive casewill be discarded in the subsequent transport analysis.

11

B. Optimization of step 1 of the transport process

Firstly, we computed the ground state of the left wellby using the imaginary time propagation technique. Westart by considering 3N atoms in such a way that we haveprecisely N atoms in each well. This approach is validwhen the three wells are far apart. The imaginary timepropagation for 3N atoms yields a wave function whichis the sum of three inverted parabolas, for large N , oralmost Gaussian functions, for small N . Each of thesethree spatially separated density profiles is localized inone of the three wells. Then, we selected the part of thewave function localized in the left well, properly normal-ized, as initial state. This new wave function describesprecisely a BEC with N atoms [we have checked the sta-tionarity of the solution by propagating it in real time viaGPE with N in the mean-field potential and as confine-ment the initial (static) potential]. When the two outertraps move towards the middle one, since the Gaussianpotentials do overlap, the three wells cannot be treated asindependent anymore (see also Fig. 1 lower panel). Eventhough the condensate wave functions of each of the threewells do not overlap, the curvature of the outer wells isdifferent from the middle one. Hence, it is no longerstraightforward to determine how many atoms are con-tained in each well. To overcome this problem, we propa-gate adiabatically the condensate wave function trappedin the well centered in x′0 towards the one centered inx0. The trap position x0 is chosen as the minimum sep-aration between the three wells such that N |ψ(x, T1)|2 islocalized only on the left well, that is, the atomic densityin the middle and right wells can be effectively neglected.We underscore that the value of x0 crucially depends onN , for fixed g1D. Hence, the adiabatically evolved con-densate wave function is chosen as goal state of step 1of the transport process. We then analyzed how fast theinitial state can be propagated towards the goal state, bysimulating the dynamics for different transport times T1,which have been chosen much smaller than the adiabatictransfer time.

We have considered atomic ensembles with N = 10 or200 87Rb atoms per well (the latter have been recentlyobtained in experiments [51]). For 10 atoms the initialpotential configuration is the same as for the single par-ticle scenario (Fig. 1 top), with x0 = −3.0µm and aslightly smaller trap frequency ωx = 2π × 702 Hz withrespect to 85Rb. For N = 200, since the condensatewave function has a much larger width than the singleparticle one, the dipole potential has to be adjusted. Tothis aim, in order to keep the (initial) lattice periodic-ity fixed, that is, |x′0| = 6.5µm (i.e., with laser beamwaist 1.3 µm), the potential depth has been increased upto V0 = kB × 25µK. Such a potential depth implies asingle-well trap frequency ωx = (4V0/mσ

2)1/2 ≃ 2π 12kHz, very similar to the trap geometry of Ref. [51], andx0 = −3.8µm as minimal (target) separation.

4 6 8 10 12 14 16 18 20 2210

−4

10−2

100

I

0.1 0.3 0.5 0.7 0.9 1.1 1.310

−4

10−2

100

I

T1 [ms]

Figure 11. (Color online). Overlap infidelity vs. transporttime: black (solid) line with the control pulse Dho(t) definedin Eq. (22); the other two lines show the infidelity obtainedwith the CRAB optimized control pulse dopt(t) = Dho(t)gopt(t)for Ng = 8 (dashed line) and Ng = 16 (dashdot line). Theupper panel corresponds to N = 10, whereas the lower one toN = 200.

As initial guess for the control pulse we used

Dho(t) =

υ2mt2

∆x 0 ≤ t < ∆x2υm ,υmt− ∆x4 ∆x2υm ≤ t <

∆xυm

υ2m(T1−t)2

2(∆x−υmT1)+∆x ∆xυm ≤ t ≤ T1

,(22)

where ∆x = |x′0| − |x0|, and υm = 3∆x/2T1 is the maxi-mum trap velocity during the transport. Such a controlpulse has been proven to be optimal for a 1D condensatein a moving harmonic potential at the transport timesT1,n = 3(2n+1)π/ωx with n ∈ N [36]. Thus, there existsa minimum transport time, T1,0 = 3π/ωx, for which noexcitation in the condensate is produced.In exactly the same way as for the single-atom trans-

port, we investigated the (quantum) speed limit of step1 of the transport process, whose results are illustratedin Fig. 11 for N = 10 (top) and N = 200 (bottom)atoms with repulsive interaction. For 200 atoms the po-tential depth is about V0 ≃ 43.5~ωx, the chemical poten-tial µ = 39.2~ωx whereas in the Thomas-Fermi limit wehave µTF = 37.4~ωx [69]. Thus, the system is well withinthis limit. Concerning the optimization, the CRAB al-gorithm works precisely as we described in Sec. III A,with the only difference that we have to substitute theSchrödinger equation with the GPE and define the con-trol pulse as d(t) = Dho(t)g(t), where g(t) is given byEq. (4). Besides, the overlap infidelity is defined againthrough Eq. (2), where ψ′g(x) ≡ ψ(x, Tad) is the stateobtained adiabatically, Tad ∼ 3 ms, starting from the

12

ground state of the left well of the initial potential con-figuration with trap separation |x′0| = 6.5µm. The sameprocedure is used for N = 10 atoms.

We see from Fig. 11 that, while for N = 10 the infi-delity decreases monotonically with respect to the trans-port time T1, for N = 200 this is not the case, and itbecomes a monotonic function only for Ng = 16. Weattribute this behavior of the infidelity to a non perfectrevival of Bogoliubov excitation modes present during thetransport process, which have a larger impact for biggercondensates, because of the larger non-linear interaction.To further improve the results one could also optimizethe dynamics of the Bogoliubov collective excitations, forinstance, by solving the time-dependent Bogoliubov-deGennes equations [70]. This approach, however, wouldallow to engineer the Bogoliubov modes, but at the ex-penses of a very demanding numerical optimization.

Furthermore, as shown in Fig. 11, the transport timesfor N = 200 are shorter than in the single-particle andsmall condensate cases. This is basically due to a shortertransport distance ∆x = 2.7µm (in the single-particlecase ∆x = 4µm) and to the trap frequency, which is∼ 16 times larger than in the single atom scenario. Fur-thermore, we see that the control pulse (22) is an excel-lent guess with satisfactory overlap infidelities up to 1ms for 200 atoms and up to 16 ms for 10 atoms, evenfor transport times T1 6= T1,n. Notably, with respect tothe single particle optimization, the addition of harmon-ics does not improve significantly the overlap infidelityfor short transport times. This behavior may also berelated to the initial guess for the coefficients {Ak, Bk},for which we always started by setting their initial val-ues to zero. Indeed, this may occur also for T1 = 0.7 ms(N = 200), where the control pulse with Ng = 16 yieldsa slightly worst overlap infidelity with respect to the oneobtained with Ng = 8. The choice for the initial valuesof {Ak, Bk} might be not the right one, since the controllandscape may have several minima: the larger the num-ber Ng is, the larger the control landscape. Thus, ourinitial choice likely produces an optimal control solutiontrapped in a local minimum that is not the same for alarger Ng. We also note that by performing the opti-mization on the frequencies ωk instead of the coefficients{Ak, Bk} the improvement in the infidelity is very small.Regarding the quantum speed limit [54], it can be

roughly fixed to 0.5 ms for N = 200, which is larger than1/νx ≃ 0.08 ms and is slightly smaller than T1,0 ≃ 0.13ms. We (numerically) defined the limit by consideringthe time for which the infidelity is approximately 10−3.This is a reasonable threshold to quantify the error onthe distance between the state evolved until time T1 andthe goal state. We note that the quantum speed limitis roughly determined by maxt∈[0,T1]{h/[Eg(t)−Ee(t)]},where h is the Planck constant, Eg(t) is the instanta-neous ground state energy, and Ee is the instantaneousenergy of the first excited state. For shorter times, it isnot physically possible to bring the system in the groundstate of the trap without populating excited energy lev-

0 0.1 0.2 0.3 0.4 0.5−0.02

−0.01

0

0.01

Dho(t

)−

dopt(t

)[µ

m]

t [ms]

0 0.25 0.53

4

5

6

7

dopt(t)

[µm

]

t [ ms]

Figure 12. (Color online). Difference between the guess con-trol pulseDho(t) and the optimal one of step 1 of the transportprocess for T1 = 0.5 ms and Ng = 16 (N = 200). In the insetthe optimal control pulse is displayed.

els, which, on the other hand, are needed, during theinterval (0, T1), to perform a fast transport of the atom.Instead, for N = 10 atoms, the quantum speed limit canbe roughly fixed to T1 = 16 ms, where the infidelity isabout 10−3. Even though in the single particle scenariowe had a slightly higher value of the trap frequency, be-cause of the use of 85Rb atoms, it is interesting to notethat already a small atomic cloud significantly alters thequantum speed limit, which, for a single atom, has beenestimated around 3 ms.In Fig. 12 the difference between the initial guess

(22) and the optimal control pulse for N = 200 atoms,T1 = 0.5 ms and Ng = 16 is depicted. This plot showshow small is the correction on the guess control pulse,even though it is quite important to decrease by morethan an order of magnitude the overlap infidelity. ForN = 10 atoms, the initial guess pulse (thick line inFig. 13) is rather different with respect to the CRAB op-timized one. This larger distortion is due to the fact thatsince the potential depth V0 = 2.55~ωx and (initial) trapseparation are the same as in the single particle scenario,the potential wells are not deep enough to consider thecontrol pulse of Eq. (22), optimal for a harmonic trap, asa good transport pulse. Indeed, while the single-particleenergy is about 0.46~ωx, the chemical potential for 10atoms is µ ≃ 1.87~ωx.Finally, we also investigated the robustness of the op-

timal control pulse for N = 200 against fluctuationsof the outer trap positions like for the single atom dy-namics. For the transport time T1 = 0.5 ms the op-timal solution obtained with CRAB is rather robust:the overlap infidelity changes from 0.0012 to 0.0046 for

13

0 2 4 6 8 10 12 14 163

4

5

6

7

d(t

)[µ

m]

t [ms]

Figure 13. (Color online). Control pulses of the step 1 ofthe transport process for N = 10 atoms: initial guess givenby Eq. (22) thick (black) line, and optimal CRAB pulse withNg = 16 thin (red) line. The transport time is T1 ≃ 15.9 ms.

ashake = ℓx ≃ 0.1µm. This effect is due to the coopera-tive behavior of the atoms in the collective motion of thecondensate. Instead, we did not investigate the effect ofdimensionality, because, unlike in the single particle sce-nario, the nonlinear term appearing in the GPE is alsoaffected by the augmented space geometry, and thereforethe comparison would not be fair (apart from the issueof validity in the quasi-2D regime).

C. Optimization of step 2: SAP process

The optimization of SAP with interacting particles ismore difficult with respect to the single atom scenario.Indeed, as also discussed in Ref. [65], in the spectrumof the nonlinear Gross-Pitaevskii Hamiltonian (16) loopsnear the avoided crossing points and new eigenstates ofĤgp emerge when enhancing the nonlinear interaction.As pointed out by Graefe et al. [65], within a three-modemodel, SAP, in order to work in the nonlinear regime, hasto fulfill the following two conditions: (i) g1DN∆ ≥ 0;(ii) g1DN/ℓx < gc = ∆. Here ∆ represents a detuningbetween the three wells, that is, the resonance conditionneeded for SAP. We note that the resonance conditionin this case imposes that the onsite energies of the wells,~ωk(t) + µk(t), are constant at all times, where ωk(t) isthe local frequency of the k-th well and µk(t) the corre-sponding chemical potential at time t. The inequality (ii)shows that there exists an upper bound on the nonlinearinteraction strength for the realization of SAP. The prob-lem we are studying, however, cannot be strictly treatedwithin a three-mode approximation. Nevertheless themodel will be used as a guideline when discussing theresults of the optimization.

As for the single particle study, we applied the CRABalgorithm in order to understand whether optimal con-trol can improve the performance of the SAP protocol.

Both for N = 50 and 200, however, we noticed that fora fixed number of harmonics (Ng = 10) CRAB was notable to reduce the value of the overlap infidelity obtainedwith the initial guess control pulses (11). This (empiri-cal) observation holds both when we are optimizing thecontrol pulse by searching for the optimal set of coef-ficients Ak, Bk, and when we seek the optimal set offrequencies ωk. Moreover, we numerically noticed thatthe convergence of the algorithm to the value of the over-lap infidelity obtained with the initial guess control pulsetakes longer than in the single-atom case. Even thoughin these two cases the number of atoms is likely muchlarger than the one allowed for the realization of SAPin the nonlinear regime, we attribute the occurrence ofsuch a phenomenon to a more elaborated control land-scape topology, that is, a control landscape with a largenumber of local minima due to the emergence of neweigenstates in the system. We did not further investigatethis aspect, which deserves a deeper analysis in a sep-arated work, but we rather chose to further reduce thenumber of atoms to N = 10. In this case CRAB wasable to improve the performance of the protocol withrespect to the initial guess control pulse. As alreadypointed out in the previous section, with respect to thesingle-atom scenario, here we used 87Rb atoms which im-ply a smaller trap frequency ωx and a trap separationx0 = −3.0 µm. Apart from these small changes, dueto a different atomic species and a broader size of theatomic sample, the trap configuration is essentially theone of the single-atom case. Nevertheless, the optimiza-tion carried out for different transport times T2 could notgo below ∼20% of overlap infidelity and ∼10% of pop-ulation in the middle trap. The result of such a studyis illustrated in Fig. 14. The obtained results cannot beimproved by further optimizing the frequencies ωk. Thisshows that even though optimal control can improve theperformance of the protocol, there is however a physicallimit due to the SAP resonance condition for which nofurther optimized dynamics can be achieved. Indeed, att = T2/2 the separation between the wells is minimal andwe can roughly estimate the detuning as ∆ ≃ 0.15 ~ωx,whereas g1DN/ℓx ≃ 5.56 ~ωx, which shows how condi-tion (ii) is not satisfied even with only N = 10 atoms.To increase ∆ one should further reduce δx0, but then thethree wells merge in a single one, or, alternatively, by re-ducing the atom number. In this case, however, the BECwould be very small and the GP description might bealso questionable. Although with a different trap setup,the analysis carried out in Ref. [47] also shows that theoverlap infidelity increases quite quickly with the num-ber of atoms and that even with only two 87Rb atoms the(non-optimized) performance of SAP is quickly harmed(∼16% of infidelity). Besides, as Fig. 14 illustrates, thebehavior is not monotonic, which is probably related toa non optimal dynamics of the Bogoliubov modes.

Finally, concerning the population of the middle trap,Fig. 14 shows that it is almost constant with a mini-mum of about 0.1. We note that, in comparison with the

14

50 100 150 200 250 300 350 400 4500.1

0.3

0.5

0.7

0.9

I(T

2)

50 100 150 200 250 300 350 400 4500.05

0.1

0.15

0.2

Pc(T

2)

T2 [ms]

Figure 14. (Color online). Overlap infidelity (top) and prob-ability of occupancy of the middle trap (bottom) vs. timefor the optimization of the SAP protocol for N = 10 inter-acting 87Rb atoms. The minimum allowed trap separation isδx0 = 1.42 µm and v±1(t) = 1.

single-atom case, we did not further minimize the pop-ulation of the middle well, since the transport efficiencywas already lower, and therefore we preferred to focuson the minimization of the overlap infidelity [i.e., we setwP = 0 in Eq. (15)]. Nevertheless, the CRAB optimiza-tion has been able to further reduce the population withrespect to the one obtained with the initial guess controlpulse.

V. CONCLUSIONS

In this paper we have numerically investigated the per-formance of the SAP protocol by means of optimal con-trol both at the single particle and at the many-bodylevel. In our analysis we have considered trap param-eters, atomic species, and atom numbers that are usedin current experiments [71]. The transport process hasbeen split in three steps, because of the initial large trapseparation. The first step brings the atom(s) localized inthe left well closer to the middle well in such a way thattunneling between the three wells occurs, therefore en-abling the realization of the second step of the transport,that is, the SAP process. Afterwards, the third step ofthe transport process brings further away from the mid-dle well the atom(s) localized in the right well. We haveseen that while we can easily achieve the quantum speedlimit, both for the single particle and the condensate sce-nario, for the first and last steps of the dynamical trans-port process, the second one requires a higher degree ofcontrol already for small transport time reductions with

respect to the “adiabatic” times. In the single atom case,we observe a smaller population in the middle trap whenthe system is forced to follow the second excited state ofthe trap (i.e., time-independent v±1) rather than follow-ing the actual dark state (i.e., time-dependent v±1). Inthe latter case, due to the different energy level of themaxima of the triple well configuration, the node of thedark state wave function is not localized within the mid-dle trap, but outside. This fact forced us to additionallyengineer the shape of the dark state wave function ren-dering the control landscape more complicated. Thus,we had to make a trade-off between transfer efficiencyand suppression of the middle trap population. In addi-tion, we observed that the engineering of the dark statereduces the robustness against trap and time delay fluc-tuations of the optimal control pulse. We note that, inorder to further improve the transfer efficiency and re-duce the population of the middle trap, by engineeringproperly the potential, one could employ a programmableand computer controllable nematic liquid-crystal spatiallight modulator, where the trap separation can be variedby changing the periodicity of the modulator [72]. Alter-natively, optical superlattices can be used, which wouldallow to fix the three minima at the same energy level aswell as the two maxima.

The optimization of the SAP protocol for a condensatestrongly relies on the atom number and onsite energy ofthe wells. We have investigated in some detail the perfor-mance of the protocol forN = 10 atoms with repulsive in-teraction. The analysis showed that the CRAB algorithmis able to improve the transport efficiency with respect tothe one obtained with the initial guess control pulse, butthe maximum attainable efficiency, for a transport timenot longer than 450 ms, is about 80% with a populationin the middle well of about 10%. It is not clear whetherlonger times could yield a better efficiency, which wouldrequire a longer computational time, but if this wouldbe the case, one has also to take into account the effectsof decoherence. For instance, if we consider atom chiptechnology [4], where the expected limits due to surface-induced decoherence of motional states are comparableto the ones of the hyperfine states, which have coherencetimes of about 1s [73], our analysis already shows thatwe are actually close to the limit of the SAP protocol.This ultimate limit, for a relatively small BEC, is dueto the emergence of new eigenstates and crossing levels,as discussed in detail in Ref. [65], which break down theSAP protocol when the nonlinear interaction exceeds acritical value.

In summary, from our investigations, it emerges thatwhile at the single atom level SAP can be optimizedbelow the 0.1% level, and possibly observed in currentexperiments, the application of an optimized SAP tech-nique to a condensate is rather limited, already even withsmall number of atoms. On the other hand, it would beinteresting to investigate more precisely and more gener-ally the influence of the nonlinear interaction of BEC onthe quantum speed limit of a certain dynamical process,

15

and this will be pursued in future work.

ACKNOWLEDGMENTS

A.N. is grateful for the invitation to UniversitatAutònoma de Barcelona and thanks Tommaso Canevafor useful hints in the implementation of the CRAB al-gorithm. We acknowledge financial support from theEU Integrated Project AQUTE, QIBEC, PICC, theDeutsche Forschungsgemeinschaft within the Grant No.SFB/TRR21 (T.C.), the Marie Curie Intra EuropeanFellowship (Proposal No. 236073, OPTIQUOS) withinthe 7th European Community Framework Programme(A.N.), financial support through Spanish MICINN con-tracts FIS2008-02425 and CSD2006-00019, the CatalanGovernment contract SGR2009-00347, and Grant No.AP 2008-01275 from the Spanish MICINN FPU Program(A. B.).

APPENDIX

The determination at each time of v−1(t) and v1(t)is a rather complicated nonlinear minimization problem.In our simulations, however, we noticed that an excel-lent approximation to the values of v−1(t) and v1(t) isgiven by the following procedure: at the beginning thepositions of the minima of the trapping potential (1) aredetermined by looking for the roots {xL, xC , xR} of thefunction

V ′(x, t) =

1∑

k=−1

[x−kdk(t)] exp{

− (x− kdk(t))2

2w2

}

, (23)

where v±1 = 1. Then, we use the following formulae:

v−1(t) =

{[

exp

(

− x2C

2w2

)

− exp(

− x2L

2w2

)][

exp

(

− (xC − d1(t))2

2w2

)

− exp(

− (xR − d1(t))2

2w2

)]

−[

exp

(

− x2C

2w2

)

− exp(

− x2R

2w2

)][

exp

(

− (xC − d1(t))2

2w2

)

− exp(

− (xL − d1(t))2

2w2

)]}

/

{[

exp

(

− (xC − d1(t))2

2w2

)

− exp(

− (xL − d1(t))2

2w2

)][

exp

(

− (xC + d−1(t))2

2w2

)

− exp(

− (xR + d−1(t))2

2w2

)]

−[

exp

(

− (xC + d−1(t))2

2w2

)

− exp(

− (xL + d−1(t))2

2w2

)][

exp

(

− (xC − d1(t))2

2w2

)

− exp(

− (xR − d1(t))2

2w2

)]}

,

(24)

v1(t) =v−1(t)

[

exp(

− (xC+d−1(t))2

2w2

)

− exp(

− (xL+d−1(t))2

2w2

)]

− exp(

− x2C

2w2

)

− exp(

− x2L

2w2

)

exp(

− (xC−d1(t))22w2)

− exp(

− (xL−d1(t))22w2) (25)

These solutions are obtained by solving the system of lin-ear equations: V (xL, t) = V (xC , t), V (xR, t) = V (xC , t).Finally, we also mention that our numerical simula-

tions of both the Schrödinger and the Gross-Pitaevskii

equation have been performed by means of the split op-erator technique together with the fast Fourier transformalgorithm [63].

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